For each value of $x,$ $f(x)$ is defined to be the minimum value of the three numbers $2x + 2,$ $\frac{1}{2} x + 1,$ and $-\frac{3}{4} x + 7.$  Find the maximum value of $f(x).$
Solution: We plot the lines $y = 2x + 2,$ $y = \frac{1}{2} x + 1,$ and $y = -\frac{3}{4} x + 7.$

[asy]
unitsize(0.5 cm);

real a, b;

a = -3;
b = 8;

draw((a,2*a + 2)--(b,2*b + 2));
draw((a,a/2 + 1)--(b,b/2 + 1));
draw((a,-3/4*a + 7)--(b,-3/4*b + 7));
draw((a,2*a + 2)--(-2/3,2/3)--(24/5,17/5)--(b,-3/4*b + 7),linewidth(1.5*bp) + red);

label("$y = 2x + 2$", (b,2*b + 2), E);
label("$y = \frac{1}{2} x + 1$", (b,b/2 + 1), E);
label("$y = -\frac{3}{4} x + 7$", (b,-3/4*b + 7), E);
label("$y = f(x)$", (0,-2), red);
label("$(-\frac{2}{3}, \frac{2}{3})$", (-2/3, 2/3), NW);
label("$(\frac{24}{5}, \frac{17}{5})$", (24/5, 17/5), N);
[/asy]

The intersection of lines $y = 2x + 2$ and $y = \frac{1}{2} x + 1$ is $\left( -\frac{2}{3}, \frac{2}{3} \right),$ and the intersection of lines $y = \frac{1}{2} x + 1$ and $y = -\frac{3}{4} x + 7$ is $\left( \frac{24}{5}, \frac{17}{5} \right).$

We can show that $f(x)$ is increasing on the interval $\left( -\infty, \frac{24}{5} \right],$ and decreasing on the interval $\left[ \frac{24}{5}, \infty \right).$  Thus, the maximum value of $f(x)$ is $f \left( \frac{24}{5} \right) = \boxed{\frac{17}{5}}.$